The minimum generator sequence [1]} for permutation \(p\)  of a permutation group \(G\)  using the generating set \(S\)  is a minimum-length sequence consisting of generators in \(S\)  whose composition is \(p\) . For example, permutation \(\pi =4\,2\,3\,1 \in \mathfrak {S}_4\)  can be expressed as \(\pi =(1\,2)(2\,3)(3\,4)(2\,3)(1\,2)\)  of length 5 or \(\pi =(1\,4)\)  of length 1 using the generating set \(S_6^\prime =\lbrace (i\,i+1):1 \le i < 4\rbrace  \cup \lbrace (1\,4)\rbrace \)  in which \(\pi =(1\,4)\)  is the minimum generator sequence of length 1. Further, if an element \(s\)  in the generating set \(S\)  can be expressed as the product of elements in \(S\)  other than \(s\) , then the generating set is called redundant [2]}. For example, \(S_6\)  is redundant, since \((1\,n)\)  can be expressed as \((1\,2)(2\,3)\cdots (n-1\,n)\cdots (2\,3)(1\,2)\) .
